[MUSIC] Hello. So we have been exploring the space of our dimensionless parameters, to see if some approximation allowed building simple enough models. This approach has been fruitful in the limit of very low reduced velocities. The fluid could then be considered as still, with regard to the dynamics of the solid. As a consequence, and in the limit of small motion of the solid, we obtain very general results on the form of the interaction force exerted by the fluid on the solid. We found fluid-added stiffness, fluid-added mass, fluid-added damping, more complex history effects of viscosity, and even coupling with fluid modes. These models are quite useful in practice, in off-shore engineering, in ship design, or biomechanics, just to cite a few. But this limit situation clearly does not cover all possible interactions. Let us now explore the effect of flow. [MUSIC] Look at the case of vibration of a plane tail as illustrated here. The fluid velocity across the wing is certainly not small. Let us see if we can build some model for this, and many other phenomena that would be related to the effect of flow. [MUSIC] Conversely to the previous approximation, and quite symmetrically, in fact, we now explore the case where the reduced velocity is very large. Remember that the reduced velocity is the ratio of the two time scales, fluid and solid. A very large reduced velocity means that the time scale of the solid dynamics is much longer than that of the fluid dynamics. In other terms, the time of oscillation or of wave propagation in the solid is now much longer than the time of convection of a free particle across the length scale L. What does it mean now? Let us imagine now that we follow in time the evolution of a quantity pertaining to the solid. For instance, the displacement of a material point. This quantity evolves with a long time scale that I call T SOLID. Conversely, a quantity pertaining to the fluid dynamics, such as the position of a fluid particle, evolves on a much shorter time scale, T FLUID. Intuitively, we may imagine that there is a limit when these time scales are very different, such that the dynamics of the fluid occurs in interaction with a fixed solid. This would certainly a be much more simple framework to model fluid-solid interactions if the solid does not seem to move. Note that by stating that the solid is fixed, I do not mean that it does not move. I just say that its own motion occurs so slowly that I can neglect it when I'm computing the motion of the fluid. But of course I will also have to compute it at some point. So we are looking for the framework of an approximation that is exactly the reverse of that of the previous lectures. Instead of a non-moving fluid, it is a non-moving solid. As an example, let us consider the oscillating wing that I showed before. The flow velocity in the reference frame of the wing is of the order of a hundred meters per second, say. So that is T fluid is the order of .01 second for wing of about a meter of chord. Then one oscillation of the wing occurs in about one second. Consequently the reduced velocity, which is a ratio between one second and .01 second, is about 100. Which is not very large, but might be large enough to do something. [MUSIC] All I've just said here was essentially an intuition that if the two time scales are so separated, there is going to be a more solvable problem. Let us see that in the equations themselves. The first step is to have a good choice of the dimensionless numbers. The ones we choose originally where the Reynolds number, the Froude number, and the Cauchy number. They're all based on the scalar velocity U naught. This is perfectly adequate. And there is no need to change them this time. [MUSIC] In our equations it is more appropriate to use the dimensionless time based on the fluid dynamics, not on the solid dynamics which is so slow. So we have t tilde equal t over T FLUID where T FLUID is L over U naught, A time scale of 1 in t tilde is appropriate for the description of the dynamics of the fluid. Finally, we shall use dimensionless velocities and pressures based on U naught. Mainly U tilde equal U over U naught. And P tilde equal P over Rho U naught squared. [MUSIC] Now, let us write down the dimensionless equations using these variables. To do this, we only need to replace the dimensional variables by the dimensionless ones in the original dimensional equations. We have already done this kind of change of variable in the previous part of the course, when we explored the case of a very low reduced velocity. So for the fluid we have the mass balance and the momentum balance as before. On the solid side, I just state here that the displacement can be represented using a single mode approximation. For this mode, I use the dimensionless oscillator equation. At the interface, I write the kinematic condition and the dynamic condition in dimensionless forms. They state that the velocities are continuous at the interface, for the first one. And that the force that applies on the mode f sub FS is the projection of the local fluid loadings on the modal shape Phi. Now that we have these equations, we should be able, somehow, to see how our assumption of a very high reduced velocity would simplify the equations. [MUSIC] Again, the reduced velocity is going to appear in the scaling of the boundary conditions on the fluid domain. The scalar U naught corresponds to the reference velocity of the fluid, which I can symbolically write as a boundary condition on the fluid. The velocity there scales as U naught so that in the dimensionless value it is of the order of U naught over U naught, which is 1. At the fluid solid interface the kinematic condition is that the fluid velocity is equal to the solid velocity. What is the magnitude of the solid velocity? We know that the solid displacement scales as Ksi naught and that it evolves in a time scale T solid. This means that the dimensionless displacements scale as Ksi naught over L. Which is exactly the displacement number D. This displacement evolves in a dimensionless timescale of T SOLID over T FLUID, which is precisely the reduced velocity, UR. So to summarize, Ksi bar is of order D and it varies of a time scale of UR. Then we can say that the order of d Ksi bar over dt bar is D over UR. So the order of the velocity in the fluid at the interface is also D over UR. [MUSIC] Let us summarize. The dynamics of the fluid is governed, on one hand, by a condition of order one and on the other end by a condition of order D over UR as interface. Certainly if UR is much larger than D the second condition can be set to zero without changing much the result. This correspond exactly to what we meant by neglecting the solid dynamics. The solid moves so slowly that its velocity is not expected to play any role in the fluid dynamics. [MUSIC] In the general case, we have an interface between the fluid and the solid, which has a deformation and a velocity. In the approximation that we just built it has deformation but no velocity. This defines the quasi-static aeroelasticity approximation. The motion of the solid is, from the point of view from the fluid, Quasi Static. We shall say that the position of the interface is frozen in time. [MUSIC] So, in the general case, the fluid dynamics and the solid dynamics are coupled by the kinematic and dynamic conditions, at interface, and they evolve simultaneously. But in our approximation we have two dynamics. One slow, which is that of the solid, and one fast, which is that of the fluid. The solid dynamics gives the position of the interface through a kinematic condition that is considered as time independent for the fluid dynamics. This means that we're back to classical problem of fluid dynamics with a rigid boundary. And we know a lot about that. We have plenty of results and methods from theory, experiments, computations. Then the solution of the fluid dynamics gives a load at the interface. With that we can compute a solid dynamics and so on. So the two dynamics are coupled, but because we have a separation of time scales, we do not have to solve them simultaneously, and this is a major result. In other terms, if we go back to our example of a wing oscillating in a flow this means that we have broken the problem of the interaction as a succession of problems of flow around a fixed deformed wing plus the problem of vibration. This seems much simpler. And it is. To summarize, we have found that for very large reduced velocities, we can build a model where the velocity of the solid at the interface can be neglected. And the key result is that we can use all we know in fluid mechanics with rigid boundaries to predict the motion of the interface. Next we shall see if such an approximation allows predicting flow induced motions such as the example we showed. [MUSIC]
